Local convergence of an optimal method of order four for solving non-linear system. (English) Zbl 1497.65093
Summary: In this article, we consider a method of order four which is optimal for solving a non-linear system in the Banach space domain. The local convergence analysis is provided by developing hypotheses based only on derivative of order one and \(\omega \)-continuity conditions, whereas there was no such analysis done by the authors who proposed this method. In fact, they have assumed hypotheses up to fifth derivative for convergence. However, derivatives of order higher than one are not found in the method. Hence, the usage of this method is restricted to equations with at least five times differentiable operators. Further, radii of domain of convergence, error estimates and results on uniqueness of the solution based on Lipschitz type conditions which were not available before are provided. By carrying out this type of analysis, the working range of the method is expanded and suitable numerical illustrations are discussed to complete this study.
MSC:
| 65J15 | Numerical solutions to equations with nonlinear operators |
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